I think you'll find that references something called a "Wallet-Size" birth certificate available from some states.
Here's a link (pdf file) to a Connecticut birth certificate app, which includes an option for the "Wallet-Size," with this caveat:
NOTE THAT THE WALLET SIZE BIRTH CERTIFICATE CONTAINS LESS INFORMATION THAN THE FULL SIZE CERTIFICATE. IT MAY NOT SATISFY ALL PROOF OF IDENTIFICATION REQUIREMENTS SUCH AS THOSE NEEDED TO OBTAIN PASSPORTS.
Sorry for the caps -- that's directly from the app.
The certification of live birth -- (I have one of those from another state. It is very similar to Obama's and it was acceptable to obtain a passport.) -- contains all the requirements you quoted: "registrar's raised, embossed, impressed or multicolored seal, registrar’s signature, and the date the certificate was filed with the registrar's office, which must be within 1 year of your birth."
Barry’s Certification of Live Birth is a short form - Abstract.
Whether wallet-sized, billboard-sized, or postage stamp-sized, it’s still a short form abstract and according to the Department of State website “may not be acceptable for passport purposes”.